Question

A manufacturer estimates that each unit of a particular commodity can be sold for $3 more than it costs to produce. There is also a fixed cost of $17000 associated with the production.
(A) Express total profit P(x) as a function of the level of production x.
(B) How much profit (or loss) is there when x = 20,000 units are produced? When 5,000 units are produced?

Answer 1

The profit made as a function of x ;

• P(x) = 3x - 17000
• When x = 20,000 ; P(x) = $43,000
• x = 5000 ; P(x) = -$2000

Let :

Number of units produced = x

Profit per unit = $3

Fixed cost = $17,000

Total profit made : Revenue - Cost of production

Profit, P(x) = (profit per unit × number of units) - Fixed cost

Profit, P(x) = 3x - 17,000

B.)

When x = 20,000

P(20000) = 3(20000) - 17,000

Profit = 60000 - 17,000

Profit = 43,000

When x = 5000

P(5000) = 3(5000) - 17,000

Profit = 15000 - 17,000

Profit = - 2000

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Answer 2

The total profit as a function of x is P(x) = 3x - 17,000

(A) The profit is $43,000

(B) The loss is $2,000

The given parameters;

• fixed cost of production, = $17000

• number of productions, = x

• let the cost of each commodity = y

• projected selling price = y + 3

The following equations will be set up as follows;

Total cost price = 17,000 + x(y)

Total selling price = x(y + 3)

The total profit as a function of x is calculated as;

total Profit = total selling price - total cost price

total Profit = x(y+3) - 17,000 - xy

total Profit = xy + 3x - 17,000 - xy

total Profit = 3x - 17,000

P(x) = 3x - 17,000

(A) The profit or loss when x = 20,000 units are produced;

P(20,000) = 3(20,000) - 17,000

P(20,000) = 60,000 - 17,000

P(20,000) = $43,000

The profit is $43,000

(B) The profit or loss when x = 5,000 units are produced;

P(5,000) = 3(5,000) - 17,000

P(5,000) = 15,000 - 17,000

P(5,000) = - $2,000

The loss is $2,000

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